When is non-trivial estimation possible for graphons and stochastic block models?‡

When is non-trivial estimation possible for graphons and stochastic block models?‡ Abstract Block graphons (also called stochastic block models) are an important and widely studied class of models for random networks. We provide a lower bound on the accuracy of estimators for block graphons with a large number of blocks. We show that, given only the number $$k$$ of blocks and an upper bound $$\rho$$ on the values (connection probabilities) of the graphon, every estimator incurs error $${\it{\Omega}}\left(\min\left(\rho, \sqrt{\frac{\rho k^2}{n^2}}\right)\right)$$ in the $$\delta_2$$ metric with constant probability for at least some graphons. In particular, our bound rules out any non-trivial estimation (that is, with $$\delta_2$$ error substantially less than $$\rho$$) when $$k\geq n\sqrt{\rho}$$. Combined with previous upper and lower bounds, our results characterize, up to logarithmic terms, the accuracy of graphon estimation in the $$\delta_2$$ metric. A similar lower bound to ours was obtained independently by Klopp et al. 1. Introduction Networks and graphs arise as natural modelling tools in many areas of science. In many settings, particularly in social networks, networks display some type of community structure. In these settings, one may consider the nodes of the graph as belonging to one of $$k$$ communities, and two nodes are connected with a probability that depends on the communities they belong to. This type of structure is captured in the $$k$$-block graphon model, also known as stochastic block models. The more communities we allow in the model (or ‘types’ of nodes we consider), the richer the model becomes and the better we can hope to describe the real world. One can think of a general graphon model as an $$\infty$$-block graphon, where each node is given a label in $$[0,1]$$ rather than $$\{1, \ldots, k\}$$. Given an observed network, graphon estimation is the problem of recovering the graphon model from which the graph was drawn. In this article, we are concerned with the fundamental limits of graphon estimation for block graphons. That is, given an $$n$$-node network that was generated from a $$k$$-block graphon, how accurately can you recover the graphon? We consider the ‘non-parametric’ setting, where $$k$$ may depend on $$n$$. Our lower bounds apply even to estimation algorithms that know the true number of blocks $$k$$, though this quantity typically needs to be estimated. In many real-world networks, the average degree of the network is small compared with the number of nodes in the network. Graphons whose expected average degree is linear in $$n$$ are called dense, while graphons whose expected average degree is sublinear in $$n$$ are referred to as sparse. In this work, we prove a new lower bound for graphon estimation for sparse networks. In particular, our results rule out non-trivial estimation for very sparse networks (roughly, where $$\rho = O(k^2/n^2)$$). An estimator is non-trivial if its expected error is significantly better than an estimator that ignores the input and always outputs the same model. It follows from recent work [1–3] that non-trivial estimation is impossible when $$\rho=O(1/n)$$. Ours is the first lower bound that rules out non-trivial graphon estimation for large $$k$$. Previous work by Klopp et al. [4] provides other lower bounds on graphon estimation that are tight in several regimes. In recent work [5] that is concurrent to ours, the same authors provide a similar bound to the one presented here. Block graphon models were introduced by Hoff et al. [6] under the name latent position graphs. Graphons play an important role in the theory of graph limits (see [7] for a survey) and the connection between the graph model, and convergent graph sequences has been studied in both the dense and the sparse settings [8–11]. Estimation for stochastic block models with a fixed number of blocks was introduced by Bickel and Chen [12], while the first estimation of the general model was proposed by Bickel et al. [13]. Since then, many graphon estimation methods, with an array of assumptions on the graphon, have been proposed [14–24]. Gao et al. [19] provide the best known upper bounds in the dense setting, while Wolfe and Olhede [23], Borgs et al. [25] and Klopp et al. [4] give upper bounds for the sparse case. 1.1 Graphons Definition 1 (Bounded graphons and $$W$$-random graphs) A (bounded) graphon$$W$$ is a symmetric, measurable function $$W:[0,1]^2\to [0,1]$$. Here, symmetric means that $$W(x,y)=W(y,x)$$ for all $$(x,y)\in [0,1]^2$$. For any integer $$n$$, a graphon $$W$$ defines a distribution on graphs on $$n$$ vertices as follows: Firstly, select $$n$$ labels $$\ell_1, \ldots, \ell_n$$ uniformly and independently from $$[0,1]$$ and form an $$n\times n$$ matrix $$H$$, where $$H_{ij} = W(\ell_i, \ell_j)$$. We obtain an unlabelled, undirected graph $$G$$ by connecting the $$i$$th and $$j$$th nodes with probability $$H_{ij}$$ independently for each $$(i,j)$$. The resulting random variable is called a $$W$$-random graph and denoted $$G_n(W)$$. For $$\rho \geq 0$$, we say a graphon is $$\rho$$-bounded if $$W$$ takes values in $$[0,\rho]$$ (that is, $$\|W\|_\infty \leq \rho$$). We denote the set of graphs with $$n$$ nodes by $$\mathcal{G}_n$$, the set of graphons by $$\mathcal{W}$$ and the set of $$\rho$$-bounded graphons by $$\mathcal{W}_\rho$$. If $$W$$ is $$\rho$$-bounded, then the expected number of edges in $$G_n(W)$$ is at most $$\rho\binom n 2 = O(\rho n^2)$$. In the case that $$\rho$$ depends on $$n$$ and $$\lim_{n\to\infty}\rho\to0$$, we obtain a sparse graphon. We consider the estimation problem: given parameters $$n$$ and $$\rho$$, as well as a graph $$G\sim G_n(W)$$ generated from an unknown $$\rho$$-bounded graphon $$W$$, how well can we estimate $$W$$? A natural goal is to design estimators that produce a graphon $$\hat W$$ that is close to $$W$$ in a metric such as $$L_2$$. This is not possible, because there are many graphons that are far apart in $$L_2$$, but that generate the same probability distribution on graphs. If there exists a measure-preserving map $$\phi:[0,1]\to[0,1]$$, such that $$W(\phi(x), \phi(y))=W'(x,y)$$ for all $$x,y\in[0,1]$$, then $$G_n(W)$$ and $$G_n(W')$$ are identically distributed. The converse is true if we instead only require $$W(\phi(x), \phi(y))=W'(x,y)$$ almost everywhere. Thus, we wish to say that $$\hat W$$ approaches the class of graphons that generate $$G_n(W)$$. To this end, we use the following metric on the set of graphons, \begin{equation} \delta_2(W,W') = \inf_{{\phi:[0,1]\to[0,1] \atop \text{measure-preserving}}} {\left\| {{W_{\phi}-W'}} \right\|_2}\!, \end{equation} (1.1) where $$W_{\phi}(x,y)=W(\phi(x),\phi(y))$$ and $$\phi$$ ranges over all measurable, measure-preserving maps. Two graphons $$W$$ and $$W'$$ generate the same probability distribution on the set of graphs if and only if $$\delta_2(W, W')=0$$ (see Lovász [7], for example). Existing upper bounds for graphon estimation are based on algorithms that produce graphons of a particular form, namely block graphons, also called stochastic block models (even when it is not known that the true graphon is a block graphon). Definition 2 ($$k$$-block graphon (stochastic block models)) For $$k\in\mathbb{N}$$, a graphon is a $$k$$-block graphon if there exists a partition of $$[0,1]$$ into $$k$$ measurable sets $$I_1, \ldots, I_k$$, such that $$W$$ is constant on $$I_i\times I_j$$ for all $$i$$ and $$j$$. We can associate a graphon of this form to every square matrix. Given a $$k\times k$$ symmetric matrix $$M$$, let $$W[M]$$ denote the $$k$$-block graphon with blocks $$I_i=\big(\frac{i-1}{k}, \frac{i}{k}\big]$$ that takes the value $$M_{ij}$$ on $$I_i\times I_j$$. 1.2 Main result We are concerned with the problem of estimating a graphon, $$W$$, given a graph sampled from $$G_n(W)$$. A graphon estimator is a function $$\hat{W}: \mathcal{G}_n \to \mathcal{W}$$ that takes as input an $$n$$ node graph, that is generated according to $$W$$, and attempts to output a graphon that is close to $$W$$. The main contribution of this article is the development of the lower bound \begin{equation} \inf_{\hat{W}}\sup_{W}\underset{G\sim G_n(W)}{\mathbb{E}}[\delta_2(\hat{W}(G), W)]\ge{\it{\Omega}}\left(\min\left(\rho, \sqrt{\frac{\rho k^2}{n^2}}\right)\right)\!. \end{equation} (1.2) Combined with previous work, we can give the following lower bound on the error of graphon estimators. Theorem 3 For any positive integer $$2\le k\le n$$ and $$0< \rho\le 1$$, \begin{equation} \inf_{\hat{W}}\sup_{W}\underset{G\sim G_n(W)}{\mathbb{E}}\left[\delta_2(\hat{W}(G), W)\right]\ge {\it{\Omega}}\left(\min\left(\rho, \;\rho\sqrt[4]{\frac{k}{n}}+\sqrt{\frac{\rho k^2}{n^2}}\right)\right)\!, \end{equation} (1.3) where $$\inf_{\hat{W}}$$ is the infimum over all estimators $$\hat{W}:G_n\to\mathcal{G}$$ and $$\sup_W$$ is the supremum over all $$k$$-block, $$\rho$$-bounded graphons. If $$\rho n$$ is non-decreasing and there exists a constant $$c>0$$, such that $$\rho n\ge c$$, then $$\inf_{\hat{W}}\sup_{W}\underset{G\sim G_n(W)}{\mathbb{E}}\left[\delta_2(\hat{W}(G), W)\right]\ge {\it{\Omega}}\left(\min\left(\rho, \;\rho\sqrt[4]{\frac{k}{n}}+\sqrt{\frac{\rho k^2}{n^2}}+\sqrt{\frac{\rho}{n}}\right)\right)\!.$$ Note that $$k$$ and $$\rho$$ may depend on $$n$$. That is, the theorem holds if we consider sequences $$\rho_n$$ and $$k_n$$. Our result improves on previously known results when $$\rho=O\left(\left(\frac{k}{n}\right)^{3/2}\right)$$—that is, when the graphs produced by the graphon are sparse. The upper bound \begin{equation}\label{Kloppupper} \inf_{\hat{W}}\sup_{W}\underset{G\sim G_n(W)}{\mathbb{E}}\left[\delta_2(\hat{W}(G), W)\right]\le O\left(\min\left(\rho, \;\rho\sqrt[4]{\frac{k}{n}}+\sqrt{\frac{\rho k^2}{n^2}}+\sqrt{\frac{\rho \log k}{n}}\right)\right) \end{equation} (1.4) by Klopp et al. [4] implies that our lower bound is almost tight. In particular, if $$k$$ is constant and $$\rho$$ is within the designated range then the lower bound in Theorem 3 is tight. When $$\rho=O\left(\frac{k^2}{n^2}\right)$$, Theorem 3 implies that the error is $${\it{\Omega}}(\rho)$$, which is the error achieved by the trivial estimator $$\hat{W}=0$$. That is, in the sparse setting, the trivial estimator achieves the optimal error. To the authors’ knowledge, this is the first result that completely rules out non-trivial estimation in the case where $$k$$ is large. Recent concurrent work ([5]) provides similar bounds. The bound \begin{equation}\label{Klopplower} \inf_{\hat{W}}\sup_{W}\underset{G\sim G_n(W)}{\mathbb{E}}[\delta_2(\hat{W}, W)]\ge {\it{\Omega}}\left(\rho\sqrt[4]{\frac{k}{n}}\right) \end{equation} (1.5) is due to previous work of Klopp et al. [4] and the bound \begin{equation}\label{Neemanlower} \inf_{\hat{W}}\sup_{W}\underset{G\sim G_n(W)}{\mathbb{E}}[\delta_2(\hat{W}, W)]\ge{\it{\Omega}}\left(\min\left(\rho, \sqrt{\frac{\rho}{n}}\right)\right) \end{equation} (1.6) for constant $$k$$ follows from the results of Mossel et al. [1] and Banerjee [26]. We give details on how to derive (1.6) from their results in the Appendix. 1.3 Techniques: combinatorial lower bounds for $$\delta_p$$ Our proof of the main theorem will involve Fano’s lemma. As such, during the course of the proof, we will need to lower bound the packing number, with respect to $$\delta_2$$, of a large set of $$k$$-block graphons. While easily upper bounded, little is known about lower bounds on $$\delta_2$$. To the authors’ knowledge, this work gives the first lower bound for the packing number of $$\mathcal{W}_\rho$$ with respect to $$\delta_2$$. We will also give a combinatorial lower bound for the $$\delta_2$$ metric that is easier to handle than the metric itself. To understand our technical contributions, it helps to first understand a problem related to graphon estimation, namely that of estimating the matrix of probabilities $$H$$. Existing algorithms for graphon estimation are generally analysed in two phases: firstly, one shows that the estimator $$\hat W$$ is close to the matrix $$H$$ (in an appropriate version of the $$\delta_2$$ metric) and then uses (high probability) bounds on $$\delta_2(W,W[H])$$ to conclude that $$\hat{W}$$ is close to $$W$$. Klopp et al. [4] show tight upper and lower bounds on estimation of $$H$$. One can think of our lower bound as showing that the lower bounds on estimation of $$H$$ can be transferred to the problem of estimating $$W$$. The main technical difficulty lies in showing that a given pair of matrices $$A,B$$ lead to graphons that are far apart in the $$\delta_2$$ metric. Even if $$A,B$$ are far apart in, say, $$\ell_2$$, they may lead to graphons that are close in $$\delta_2$$. For consistency with the graphon formalism, we normalize the $$\ell_2$$ metric on $$k\times k$$ matrices, so that it agrees with the $$L_2$$ metric on the corresponding graphons. For a $$k\times k$$ matrix $$A$$, \begin{equation} {\left\| {A} \right\|_2} {\overset{\text{def}}{=}} \Big(\frac 1 {k^2} \sum_{i,j\in [k]} A_{ij}^2\Big)^{1/2} = {\left\| {{W[A]}} \right\|_2}\!. \end{equation} (1.7) As an example of the discrepancy between the $$\ell_2$$ and $$\delta_2$$ metrics, consider the matrices $$A = \begin{pmatrix} 1 & 0 & 1\\ 0&1&0\\1&0&1\end{pmatrix} \hspace{0.5in}\text{and}\hspace{0.5in} B= \begin{pmatrix} 1 & 1 & 0\\ 1&1&0\\0&0&1\end{pmatrix}.$$ The matrices $$A$$ and $$B$$ have positive distance in the $$\ell_2$$ metric, $${\left\| {{A-B}} \right\|_2} = \frac{2}{3}$$, but $$\delta_2(W[A], W[B])=0$$. One can get an upper bound on $$\delta_2(W[A],W[B])$$ by restricting attention in the definition of $$\delta_2$$ to functions $$\phi$$ that permute the blocks $$I_i$$. This leads to the following metric on $$k\times k$$ matrices, which minimizes over permutations of the rows and columns of one of the matrices: \begin{equation} \hat{\delta}_2(A,B) {\overset{\text{def}}{=}} \min_{\sigma\in\mathcal{S}_k}{\left\| {{A_{\sigma}-B}} \right\|_2}\!, \end{equation} (1.8) where $$A_\sigma$$ is the matrix with entries $$(A_\sigma)_{ij} = A_{\sigma(i),\sigma(j)}$$. This metric arises in other work (e.g. [7]), and it is well known that \begin{equation} \delta_2(W[A],W[B]) \leq \hat{\delta}_2(A,B). \end{equation} (1.9) To prove lower bounds, we consider a new metric on matrices, in which we allow the rows and columns to be permuted separately. Specifically, let \begin{equation} \hat{\hat{{\delta}}}_2(A,B) {\overset{\text{def}}{=}} \min_{\sigma, \tau\in\mathcal{S}_k}{\left\| {{A_{\sigma, \tau}-B}} \right\|_2}\!, \end{equation} (1.10) where $$A_{\sigma,\tau}$$ is the $$k\times k$$ matrix with entries $$(A_{\sigma,\tau})_{ij} = A_{\sigma(i),\tau(j)}$$. Lemma 1 (Lower bound for $$\delta_2$$) For every two $$k\times k$$ matrices $$A,B$$, \begin{equation} \hat{\hat{\delta_2(A,B)}} \leq \delta_2(W[A], W[B]). \end{equation} (1.11) Because $$\hat{\hat{\delta_2}}$$ is defined ‘combinatorially’ (that is, it involves minimization over a discrete set of size about $$2^{2k\ln k}$$, instead of over all measure-preserving injections), it is fairly easy to lower bound $$\hat{\hat{ \delta_2(A,B)}}$$ for random matrices $$A,B$$ using the union bound. In particular, it allows us to give bounds on the packing number of $$\mathcal{W}_\rho$$ with respect to the $$\delta_2$$ metric. If $$({\it{\Omega}}, d)$$ is a metric space, $$\epsilon>0$$ and $$T\subset{\it{\Omega}}$$, then we define the $$\epsilon$$-packing number of $$T$$ to be the largest number of disjoint balls of radius $$\epsilon$$ that can fit in $$T$$, denoted by $$\mathcal{M}(\epsilon, T, d)$$. The following proposition will be proved after the proof of Theorem 3. Proposition 1 There exists $$C>0$$ such that the $$C\rho$$-packing number of $$\mathcal{W}_{\rho}$$, equipped with $$\delta_2$$, is $$2^{{\it{\Omega}}(k^2)}$$, that is $$\mathcal{M}(C\rho, \mathcal{W}_{\rho}, \delta_2) = 2^{{\it{\Omega}}(k^2)}$$. Finally, we note that these techniques extend directly to the $$\delta_p$$ metric, for $$p \in [1,\infty]$$. That is, we may define $$\delta_p, \hat\delta_p$$ and $$\hat{\hat{\delta_p}}$$ analogously to the definitions above and obtain the bounds \begin{equation} \hat{\hat{\delta_p(A,B)}} \leq \delta_p(W[A], W[B])\leq \hat\delta_p(A,B), \end{equation} (1.12) along with similar lower bounds on the packing number. 1.4 Related work Work on graphon estimation falls broadly into two categories: estimating the matrix $$H$$ and estimating the graphon $$W$$. When estimating $$H$$, the aim is to produce a matrix that is close in the $$\ell_2$$ metric to the true matrix of probabilities $$H$$ that was used to generate the graph $$G$$. When estimating the graphon, our aim is to minimize the $$\delta_2$$ distance between the estimate and the true underlying graphon $$W$$ that was used to generate $$G$$. Gao et al. [19] studied the problem of estimating the matrix of probabilities $$H$$, given an instance chosen from $$W$$ when $$\rho=1$$. They proved the following minimax rate for this problem when $$W$$ is a $$k$$-block graphon: \begin{equation} \inf_{\hat{M}}\sup_{H}\underset{G\sim G_n(H)}{\mathbb{E}}\left[\frac{1}{n^2}{\left\| {{\hat{M}(G)-H}} \right\|_2}\right]\asymp \sqrt{\frac{k^2}{n^2}+\frac{\log k}{n}}, \end{equation} (1.13) where the infinimum is over all estimators $$\hat{M}$$ from $$G_n$$ to the set of symmetric $$n\times n$$ matrices, the supremum is over all probability matrices $$H$$ generated from $$k$$-block graphons. Klopp et al. [4] extended this result to the sparse case, proving that for all $$k\le n$$ and $$0<\rho\le 1$$, \begin{equation} \inf_{\hat{M}}\sup_{H}\underset{G\sim G_N(H)}{\mathbb{E}}\left[\frac{1}{n^2}{\left\| {{\hat{M}(G)-H}} \right\|_2}\right]\ge{\it{\Omega}}\left(\min\left(\sqrt{\rho\left(\frac{k^2}{n^2}+\frac{\log k}{n}\right)}, \rho\right)\right)\!, \end{equation} (1.14) where the supremum is over all probability matrices $$H$$ generated from $$k$$-block, $$\rho$$-bounded graphons. Klopp et al. [4, Corollary 3] also studied the problem of estimating the graphon $$W$$. They proved that equation (1.4) holds for any $$k$$-block, $$\rho$$-bounded graphon, $$W$$, with $$k\le n$$. They also exhibited the first lower bound (known to us) for graphon estimation using the $$\delta_2$$ metric. They proved that equation (1.5) holds for $$\rho>0$$ and $$k\le n$$. The related problems of distinguishing a graphon with $$k> 1$$ from an Erdös–Rényi model with the same average degree (called the distinguishability problem) and reconstructing the communities of a given network (called the reconstruction problem) have also been widely studied. This problem is closely related to the problem of estimating $$H$$. Recent work by Mossel et al. [1] and Neeman and Netrapalli [3] establish conditions under which a $$k$$-block graphon is mutually contiguous to the Erdös–Rényi model with the same average degree. Contiguity essentially implies that no test could ever definitely determine which of the two graphons a given sample came from. There is a large body of work on algorithmic and statistical problems in this area, and we have only cited work that is directly relevant here. 2. Lower bound for the $$\delta_2$$ metric As mentioned earlier, the main technical contribution of this article is lower bounding the $$\delta_2$$ metric by the more combinatorial $$\hat{\hat{{\delta_2}}}$$ metric. In this section, we will prove the inequality given in Lemma 1. Proposition 2 Let $$W, W'$$ be $$k$$-block graphons with blocks $$I_i=\big[\frac{i-1}{k}, \frac{i}{k}\big)$$ and $$\pi:[0,1]\to[0,1]$$ be a measure-preserving map. Then there exists a probability distribution $$\mathbb{P}$$ on $$\mathcal{S}_k$$ such that \begin{equation} {\left\| {{W_{\pi}-W'}} \right\|_2}^2 = \underset{\sigma, \tau\sim\mathbb{P}}{\mathbb{E}}\left[{\left\| {{W_{\sigma, \tau}-W'}} \right\|_2}^2\right], \end{equation} (2.1) where the expectation is taken over $$\sigma,\tau$$ selected independently according to $$\mathbb{P}$$. Proof. Let $$a_i = \frac{i-1}{k}$$ and $$p_{ij} = \mu(I_i\cap \pi^{-1}(I_j))$$. Now, consider a $$k\times k$$ matrix $$P$$ with $$P_{ij} = kp_{ij}$$. Noting that $$\sum_{j=1}^k p_{ij} = \mu(I_i) = 1/k$$ and $$\sum_{i=1}^k p_{ij} = \mu(\pi^{-1}(I_j)) = 1/k$$, we can see that $$P$$ is doubly stochastic, that is, the rows and columns of $$P$$ sum to 1. Berkhoff’s theorem states that any doubly stochastic matrix can be written as a convex combination of permutation matrices. Therefore, we have a probability distribution $$\mathbb{P}$$ on $$\mathcal{S}_k$$ such that $$P=\sum_{\sigma\in\mathcal{S}_k}\mathbb{P}(\sigma)\sigma$$ and $$\sum_{\sigma\in\mathcal{S}_k}\mathbb{P}(\sigma)=1$$ and \begin{equation} \mathbb{P}(\sigma(i)=j) = \sum\{\mathbb{P}(\sigma) \:|\: \sigma(i)=j\} = P_{ij} = kp_{ij}. \end{equation} (2.2) Taking expectations over $$\sigma,\tau$$ selected independently from $$\mathbb{P}$$, \begin{align*} \mathbb{E}\left[{\left\| {{W_{\sigma, \tau}-W'}} \right\|_2}^2\right]&=\sum_{\sigma,\tau} \mathbb{P}(\sigma)\mathbb{P}(\tau)\sum_{i,j} \frac{1}{k^2}(W(a_{\sigma(i)}, a_{\tau(j)})-W'(a_i,a_j))^2 \\ &= \sum_{i,i',j,j'} \frac{1}{k^2}\mathbb{P}(\sigma(i)=i')\mathbb{P}(\tau(j)=j')(W(a_i, a_j)-W'(a_{i'}, a_{j'}))^2 \\ &= \sum_{i,i',j,j'} p_{ii'}p_{jj'}(W(a_i, a_j)-W'(a_{i'},a_{j'}))^2\\ &= {\left\| {{W_{\pi}-W'}} \right\|_2}^2\!. \end{align*} □ Proof of Lemma 1. Proposition 2 implies that for all measure-preserving maps $$\pi:[0,1]\to[0,1]$$ and matrices $$A$$ and $$B$$ we have \begin{equation} {\left\| {{W[A]_{\pi}-W[B]}} \right\|_2}\ge\inf_{\sigma, \tau\in\mathcal{S}_k}{\left\| {{W[A]_{\sigma, \tau}-W[B]}} \right\|_2}=\inf_{\sigma, \tau\in\mathcal{S}_k}{\left\| {{A_{\sigma, \tau}-B}} \right\|_2} = \hat{\hat{{\delta_2}}}(A, B). \end{equation} (2.3) Since this is true for any $$\pi$$, we have $$\delta_2(W[A],W[B])\ge\hat{\hat{{\delta_2}}}(A,B)$$. □ 3. Proof of main theorem To prove the main theorem, we will use Fano’s lemma to find a constant that lower bounds the probability that the estimation exceeds $$\min\left(\rho, \sqrt{\frac{\rho k^2}{n^2}}\right)$$, which then implies the appropriate lower bound on the expected $$\delta_2$$ error. To that end, we aim to find a large set, $$T$$, of $$k$$-block graphons whose Kullback–Leibler (KL) diameter and $$\epsilon$$-packing number with respect to $$\delta_2$$ with $$\epsilon=\min\left(\rho, \sqrt{\frac{\rho k^2}{n^2}}\right)$$ can be bounded. Our proof is inspired by that of Gao et al. [19]. Suppose $$p, q$$ are probability distributions on the same space. Then the (KL) divergence of $$p$$ and $$q$$ is defined by $$D(p\|q) = \int\big(\log\frac{{\rm{d}}p}{{\rm{d}}q}\big){\rm{d}}p$$. For a collection $$T$$ of probability distributions, the KL diameter is defined by \begin{equation} {\rm d}_{\rm KL}(T)=\sup_{p,q\in T} D(p\| q). \end{equation} (3.1) The following version of Fano’s lemma is found in [27]. Recall that $${\log\mathcal{M}(\epsilon, T, d)}$$ is the size of the largest $$\epsilon$$-packing of $$T$$ (with distance $$d$$). Lemma 2 (Fano’s inequality) Let $$({\it{\Omega}}, d)$$ be a metric space and $$\{\mathbb{P}_{\theta} \: |\: \theta\in{\it{\Omega}}\}$$ be a collection of probability measures. For any totally bounded $$T\subset{\it{\Omega}}$$ and $$\epsilon>0$$, \begin{equation} \inf_{\hat{\theta}}\sup_{\theta\in{\it{\Omega}}}\mathbb{P}_{\theta}\left({\rm d}^2(\hat{\theta}(X), \theta)\ge \frac{\epsilon^2}{4}\right)\ge 1-\frac{{\rm d}_{\rm KL}(P_T)+1}{\log\mathcal{M}(\epsilon, T, d)}, \end{equation} (3.2) where the infimum is over all estimators and $$P_T = \{\mathbb{P}_{\theta}\;|\;\theta\in T\}$$. The following lemma gives us a way to easily upper bound the KL divergence between the distributions induced by two different graphons. Lemma 3 For any graphons $$\frac{1}{4}\le W,W'\le \frac{3}{4}$$, we have \begin{equation} D(G_n(W)\|G_n(W'))\le 8n^2{\left\| {W-W'} \right\|_2}^2\!. \end{equation} (3.3) Proof. Let $$T$$ be a variable denoting the choice of labels, so \begin{equation} \mathbb{P}_{G_n(W)}(G) = \int_{\ell\in[0,1]^n} \mathbb{P}_T(\ell)\mathbb{P}_{G_n(W)}(G|T=\ell)\,{\rm{d}}\ell. \end{equation} (3.4) Now, \begin{align*} D(G_n(W)\|G_n(W')) &= \sum_{G\in G_n} \mathbb{P}_{G_n(W)}(G)\ln\left(\frac{\mathbb{P}_{G_n(W)}(G)}{\mathbb{P}_{G_n(W')}(G)}\right)\\ &\le \sum_{G\in G_n} \int_{\ell\in[0,1]^n} \mathbb{P}_T(\ell)\mathbb{P}_{G_n(W)}(G|T=\ell)\ln\left(\frac{\mathbb{P}_{G_n(W)}(G|T=\ell)}{\mathbb{P}_{G_n(W')}(G|T=\ell)}\right){\,{\rm d}}\ell\\ &= \int_{\ell\in[0,1]^n} \mathbb{P}_T(\ell)D(\mathbb{P}_{G_n(W)}(\cdot|T=\ell))\|\mathbb{P}_{G_n(W')}(\cdot|T=\ell){\, {\rm d}}\ell, \end{align*} where the inequality follows from the log-integral inequality [28, Theorem 30.12.4]. Now, the probability density function of $$T$$ is the constant function 1, so it follows from Gao et al. [19, Proposition 4.2] that \begin{align*} D(G_n(W)\|G_n(W')) &\le 8\int_{\ell\in[0,1]^n}\sum_{i,j=1}^n (W({\ell_i},{\ell_j})- W'({\ell_i},{\ell_j}))^2{\, {\rm d}}\ell\\ &=8\sum_{i,j=1}^n \int_{\ell\in[0,1]^n} (W(\ell_i,\ell_j)- W'(\ell_i,\ell_j))^2 {\, {\rm d}}\ell\\ &\le 8n^2\int_{[0,1]^2} (W(x,y)- W'(x,y))^2{\, {\rm d}}x\,{\rm{d}}y\\ &= 8n^2{\left\| {W-W'} \right\|_2}^2\!. \end{align*} □ Recall that we are aiming to define a large set of $$k$$-block matrices that are close in KL divergence, but that are $$\epsilon$$-far apart with respect to $$\delta_2$$ (with $$\epsilon=\min(\rho, \sqrt{\frac{\rho k^2}{n^2}})$$). The following lemma shows that there exists a large set of matrices that are pairwise far in Hamming distance, even after every possible permutation of the rows and columns. We will use this in the proof of Theorem 3 to define a large class of $$k$$-block graphons that are pairwise far in the $$\hat{\hat{{\delta_2}}}$$ metric and hence the $$\delta_2$$ metric. This gives us a bound on packing number. Lemma 4 There exists a set $$S$$ of symmetric $$k\times k$$ binary matrices such that $$|S| = 2^{{\it{\Omega}}(k^2)}$$ and, for every $$B, B'\in S$$ and $$\sigma, \tau\in\mathcal{S}_k$$, we have Ham$$(B_{\sigma, \tau}, B')= {\it{\Omega}}(k^2)$$. Proof. Fix permutations $$\sigma$$ and $$\tau$$, and consider two randomly chosen symmetric binary matrices $$B, B'$$. For $$i\le j$$, let $$X_{ij}=1$$ if $$B_{\sigma(i), \tau(j)}=B'_{i,j}$$ and 0 otherwise so $$X_{ij}$$ is a Bernoulli random variable with $$\mathbb{E}[X_{ij}]=\frac{1}{2}$$. Thus, by a Chernoff bound, \begin{equation} \mathbb{P}\left(\text{Ham}(B_{\sigma, \tau}, B')<\frac{k^2}{6}\right)=\mathbb{P}\left(\sum_{i\le j}X_{ij}\le \frac{k^2}{6}\right)\le e^{\frac{-2\left(\frac{k^2}{6}-\frac{1}{2}\binom{k}{2}\right)^2}{\binom{k}{2}}}. \end{equation} (3.5) Therefore, for randomly chosen $$B, B'$$, \begin{equation} \mathbb{P}\left(\exists\sigma, \tau \text{ s.t. Ham}(B_{\sigma, \tau}, B')<\frac{k^2}{6}\right)\le e^{\frac{-2\left(\frac{k^2}{6}-\frac{1}{2}\binom{k}{2}\right)^2}{\binom{k}{2}}}(k!)^2 = 2^{-{\it{\Omega}}(k^2)}. \end{equation} (3.6) For a constant $$c>0$$, consider the process that selects $$2^{ck^2}$$ binary matrices $$\{B_i\}_{i}$$ uniformly at random uniformly at random. The probability that all pairs are at Hamming distance at least $$k^2/6$$ is at least $$1-2^{2ck^2}2^{-{\it{\Omega}}(k^2)}$$. Selecting $$c$$ sufficiently small, we get that at least one such set $$S$$ exists. □ We are not aware of an explicit construction of a large family of matrices that are far apart in $$\hat{\hat{{\delta}_2}}$$ metric; we leave such a construction as an open problem. We now proceed to the proof of Theorem 3. We will use Lemma 4 to define a set $$T$$ with packing number $$2^{{\it{\Omega}}(k^2)}$$. The elements of $$T$$ are all close in $$\|\cdot\|_{\infty}$$ norm, so using Lemma 3 we get a bound on the KL diameter. We then directly apply these bounds via Fano’s lemma. Theorem 4 For any positive integer $$k\le n$$ and $$0< \rho\le 1$$, \begin{equation} \inf_{\hat{W}}\sup_{W}\underset{G\sim G_n(W)}{\mathbb{E}}\left[\delta_2(\hat{W}(G), W)\right]\ge {\it{\Omega}}\left(\min\left(\rho, \;\sqrt{\frac{\rho k^2}{n^2}}\right)\right)\!, \end{equation} (3.7) where $$\inf_{\hat{W}}$$ is the infimum over all estimators $$\hat{W}:G_n\to\mathcal{G}$$ and $$\sup_W$$ is the supremum over all $$k$$-block, $$\rho$$-bounded graphons. Proof. Let $$S$$ be a set satisfying the conditions of Lemma 4 and let $$\eta = \min(1, \frac{k}{n\sqrt{\rho}})$$. For $$B\in S$$, define \begin{equation} Q_B = \rho\left[\frac{1}{2}\mathbf{1}+c\eta(2B-\mathbf{1})\right]\!, \end{equation} (3.8) where $$\mathbf{1}$$ is the all 1’s matrix and $$c$$ is some constant that we will choose later. That is, $$(Q_B)_{ij}=\rho[\frac{1}{2}+c\eta]$$ if $$B_{ij}=1$$ and $$(Q_B)_{ij} = \rho[\frac{1}{2}-c\eta]$$ if $$B_{ij}=0$$. Let $$T= \left\{W[Q_B] \:|\: B\in S\right\}$$. Using Lemma 3, we conclude that for all $$W, W'\in T$$, we have \begin{equation} D(G_n(W)\|G_n(W')) \le 8n^2(2c\rho\eta)^2 \le 32c^2k^2\rho, \end{equation} (3.9) so $${\rm d}_{\rm KL}(T) = O(c^2k^2\rho)$$. Let $$B, B'\in S$$ and suppose $$\sigma, \tau\in\mathcal{S}_k$$. By construction, \begin{equation} {\left\| {{({W[Q_B]})_{\sigma, \tau}-W[Q_{B'}]}} \right\|_2}^2 \ge \frac{1}{k^2}\text{Ham}(B_{\sigma, \tau}, B')(2\rho c\eta)^2 = {\it{\Omega}}(c^2\rho^2\eta^2). \end{equation} (3.10) Thus by Corollary 2, \begin{equation} \delta_2(W[Q_B], W[Q_{B'}])\ge\hat{\hat{{\delta_2}}}(W[Q_B], W[Q_{B'}])\ge {\it{\Omega}}(c\rho\eta). \end{equation} (3.11) Therefore, there exists $$D>0$$ such that if $$\epsilon = D\rho c\eta = D\min\left(c\rho, \frac{ck\sqrt{\rho}}{n}\right)$$, we have $$\log\mathcal{M}(\epsilon, T, \delta_2) = {\it{\Omega}}(k^2)$$. Fano’s lemma implies \begin{equation} \inf_{\hat{W}}\sup_{W} \text{Pr}\left(\delta_2(\hat{W}, W)\ge \frac{\epsilon}{2}\right)\ge 1-\frac{O(c^2k^2\rho)+1}{{\it{\Omega}}(k^2)}. \end{equation} (3.12) We can choose $$c$$ small enough that the right-hand side is larger than a fixed constant for all $$k$$ and $$n$$. By Markov’s inequality, we have \begin{equation} \inf_{\hat{W}}\sup_{W}\mathbb{E}\left[\delta_2(\hat{W}, W)\right] = {\it{\Omega}}\left(\epsilon\right) = {\it{\Omega}}\left(\min\left(\rho, \sqrt{\rho\frac{k^2}{n^2}}\right)\right)\!. \end{equation} (3.13) □ Proof of Proposition 1. During the course of the proof of Theorem 3, we construct $$2^{{\it{\Omega}}(k^2)}$$ graphons in $$\mathcal{W}_{\rho}$$ that are pairwise at least $${\it{\Omega}}(\rho c\eta)$$ apart in the $$\delta_2$$ distance for any $$c>0$$ such that $$|c\eta|\le\frac{1}{2}$$. Therefore, for some $$C>0$$, the $$C\rho$$-packing number of $$\mathcal{W}_{\rho}$$ is at least $$2^{{\it{\Omega}}(k^2)}$$. □ Acknowledgements We are grateful for helpful conversations with Christian Borgs, Jennifer Chayes, Olga Klopp and Alexandre Tsybakov. Funding Alfred P. Sloan Foundation; a Google Faculty Research Award; and the National Science Foundation award (IIS-1447700). A preliminary version of this work appears as ArXiv report arXiv:1604.01871 [math.ST]. This work was done while the author was visiting Pennsylvania State University. References 1. Mossel E. , Neeman J. & Sly A. ( 2014 ) Reconstruction and estimation in the planted partition model . Probab. Theory Related Fields , 162 , 431 – 461 . Google Scholar CrossRef Search ADS 2. Mossel E. , Neeman J. & Sly A. ( 2015 ) Consistency thresholds for the planted bisection model . Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC ’15 ( Rubinfeld R. ed.). New York, NY, USA : ACM , pp. 69 – 75 . 3. Banks J. , Moore C. , Neeman J. & Netrapalli P. ( 2016 ) Information-theoretic thresholds for community detection in sparse networks . Proceedings of the 29th Conference on Learning Theory, COLT ( Feldman V. Rakhlin A. & Shamir O. eds). New York, USA : JMLR , pp. 383 – 416 . 4. Klopp O. , Tsybakov A. B. & Verzelen N. ( 2015 ) Oracle inequalities for network models and sparse graphon estimation (version 1) . arXiv:1507.04118v1 . 5. Klopp O. , Tsybakov A. B. & Verzelen N. ( 2017 ) Oracle inequalities for network models and sparse graphon estimation (version 3) . Ann. Statist. , 45 , 316 – 354 . Google Scholar CrossRef Search ADS 6. Hoff P. D. , Raftery A. E. & Handcock M. S. ( 2002 ) Latent space approaches to social network analysis . J. Amer. Stat. Assoc. , 97 , 1090 – 1098 . Google Scholar CrossRef Search ADS 7. Lovász L. ( 2012 ) Large networks and graph limits . Amer. Math. Soc. Colloq. Publ. , 60 , 1 – 475 . 8. Borgs C. , Chayes J. 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Lake Tahoe, USA : Curran Associates, Inc. , pp. 692 – 700 . 17. Chan S. H. & Airoldi E. M. ( 2014 ) A consistent histogram estimator for exchangeable graph models . J. Mach. Learn. Res. Workshop Conf. Proc. , 32 , 208 – 216 . 18. Chatterjee S. ( 2015 ) Matrix estimation by universal singular value thresholding . Ann. Statist. , 43 , 177 – 214 . Google Scholar CrossRef Search ADS 19. Gao C. , Lu Y. & Zhou H. H. ( 2015 ) Rate-optimal graphon estimation . Ann. Statist. , 43 , 2624 – 2652 . Google Scholar CrossRef Search ADS 20. Latouche P. & Robin S. ( 2016 ) Variational Bayes model averaging for graphon functions and motif frequencies inference in W-graph models . Stat. Comput. , 26 , 1173 – 1185 . Google Scholar CrossRef Search ADS 21. Lloyd J. R. , Orbanz P. , Ghahramani Z. & Roy D. M. ( 2012 ) Random function priors for exchangeable arrays with applications to graphs and relational data . ( Pereira F. Burges C. J. C. Bottou L. & Weinberger K. Q. eds). Lake Tahoe, USA : Curran Associates, Inc. , pp. 998 – 1006 . 22. Tang M. , Sussman D. L. & Priebe C. E. ( 2013 ) Universally consistent vertex classification for latent positions graphs . Ann. Statist. , 41 , 1406 – 1430 . Google Scholar CrossRef Search ADS 23. Wolfe P. & Olhede S. C. ( 2013 ) Nonparametric graphon estimation . arXiv:1309.5936 . 24. Yang J. J. , Han Q. & Airoldi E. M. ( 2014 ) Nonparametric estimation and testing of exchangeable graph models . Proceedings of 17th International Conference on Artificial Intelligence and Statistics ( Kaski S. & Corander J. eds). Reykjavik, Iceland : PMLR , pp. 1060 – 1067 . 25. Borgs C. , Chayes J. T. & Smith A. ( 2015 ) Private graphon estimation for sparse graphs . Proceedings of the 28th International Conference on Neural Information Processing Systems, NIPS’15 ( Cortes C. Lee D. D. Sugiyama M. & Garnett R. eds). Cambridge, MA : MIT Press , pp. 1369 – 1377 . 26. Banerjee D. ( 2016 ) Contiguity results for planted partition models: the dense case . arXiv:1609.02854v1 . 27. Yu B. ( 1997 ) Assouad, Fano, and Le Cam . Festschrift for Lucien Le Cam ( Pollard D. Torgersen E. & Yang G. eds). New York : Springer , pp. 423 – 435 . Google Scholar CrossRef Search ADS 28. Lapidoth A. ( 2017 ) A Foundation in Digital Communication . Cambridge, UK : Cambridge University Press . Google Scholar CrossRef Search ADS Appendix. Proof of equation (1.6) We show here how to derive the lower bound in (1.6) from the results of Mossel et al. [1] and Banerjee [26]. Let $$({\it{\Omega}}_n, \mathcal{F}_n)$$ be a sequence of measurable spaces, each equipped with two probability measures, $$\mathbb{P}_n$$ and $$\mathbb{Q}_n$$. We say $$\mathbb{P}_n$$ and $$\mathbb{Q}_n$$ are mutually contiguous if for any sequence of events $$A_n$$, we have $$\lim_{n\to\infty}\mathbb{P}_n(A_n)\to0$$ if and only if $$\lim_{n\to\infty}\mathbb{Q}_n(A_n)\to0$$. Lemma A1 Let $$W_1$$ be a $$k$$-block graphon generated by a matrix with diagonal entries, $$p$$, and off-diagonal entries, $$q$$. Let $$W_2$$ be the Erdös–Rényi model with the same expected degree as $$W_1$$. If $$W_1$$ and $$W_2$$ are mutually contiguous then \begin{equation}\label{mosselbound} \inf_{\hat{W}}\sup_{W}\underset{G\sim G_n(W)}{\mathbb{E}}\left[\delta_2(\hat{W}(G), W)\right]\ge {\it{\Omega}}\left(\frac{|p-q|}{\sqrt{k}}\right)\!, \end{equation} (A.1) where $$\inf_{\hat{W}}$$ is the infimum over all estimators $$\hat{W}:G_n\to\mathcal{G}$$ and $$\sup_W$$ is the supremum over all $$k$$-block, $$\max(p,q)$$-bounded graphons. Proof. Let $$\epsilon = |p-q|$$, so $$\delta_2\left(W_1, W_2\right) = \sqrt{\frac{(k-1)^2}{k^3}\epsilon^2+\frac{(k-1)}{k^3}\epsilon^2}\ge C\frac{\epsilon}{\sqrt{k}}$$, for some constant $$C$$. Suppose, for sake of contradiction, that there exists $$\hat{W}$$ such that $$\sup_{W\in\mathcal{W}_{\rho}}\mathbb{E}_{G\sim G_n(W)}\left[\delta_2(\hat{W}(G), W)\right]$$ is not $${\it{\Omega}}\left(\frac{\epsilon}{\sqrt{k}}\right)$$. Then there exists a subsequence $$\{n_t\}_{t\in\mathbb{N}}$$ such that $$\sup_{W\in\mathcal{W}_{\rho}}\mathbb{E}_{G\sim G_{n_t}(W)}\left[\delta_2(\hat{W}(G), W)\right]\le \frac{C}{2t}\frac{\epsilon}{\sqrt{k}}$$ for all $$t\in\mathbb{N}$$. The above inequality, combined with Markov’s inequality, implies \begin{equation}\label{eqn2} \lim_{t\to\infty}Pr_{G\sim G_{n_t}(W_1)}\left[\delta_2(\hat{W}(G), W_1)\ge \frac{C}{2}\frac{\epsilon}{\sqrt{k}}\right]\to 0 \end{equation} (A.2) and \begin{equation}\label{eqn1} \lim_{t\to\infty}Pr_{G\sim G_{n_t}(W_2)}\left[\delta_2(\hat{W}(G), W_2)\ge \frac{C}{2}\frac{\epsilon}{\sqrt{k}}\right]\to 0. \end{equation} (A.3) By equation (A.3) and the contiguity of $$W_1$$ and $$W_2$$, \begin{equation}\label{eqn3} \lim_{t\to\infty}Pr_{G\sim G_{n_t}(W_1)}\left[\delta_2(\hat{W}(G), W_2)\ge \frac{C}{2}\frac{\epsilon}{\sqrt{k}}\right]\to 0. \end{equation} (A.4) Therefore, equations (A.2) and (A.4) imply that for large enough $$n$$, there exists a graph $$G$$ such that $$\delta_2(\hat{W}(G), W_1)< \frac{C}{2}\frac{\epsilon}{\sqrt{k}}$$ and $$\delta_2(\hat{W}(G), W_2)< \frac{C}{2}\frac{\epsilon}{\sqrt{k}}$$, which implies that $$\delta_2(W_1, W_2)<C\frac{\epsilon}{\sqrt{k}}$$, which is a contradiction. □ There are many results in the literature exploring when block graphons are contiguous with the corresponding Erdös–Rényi model. The following table summarizes some of the known results in this area and translates them into lower bounds on the graphon estimation problem via Lemma A1. Let $$\rho=\max(p,q)$$. © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Information and Inference: A Journal of the IMA Oxford University Press

When is non-trivial estimation possible for graphons and stochastic block models?‡

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Abstract

Abstract Block graphons (also called stochastic block models) are an important and widely studied class of models for random networks. We provide a lower bound on the accuracy of estimators for block graphons with a large number of blocks. We show that, given only the number $$k$$ of blocks and an upper bound $$\rho$$ on the values (connection probabilities) of the graphon, every estimator incurs error $${\it{\Omega}}\left(\min\left(\rho, \sqrt{\frac{\rho k^2}{n^2}}\right)\right)$$ in the $$\delta_2$$ metric with constant probability for at least some graphons. In particular, our bound rules out any non-trivial estimation (that is, with $$\delta_2$$ error substantially less than $$\rho$$) when $$k\geq n\sqrt{\rho}$$. Combined with previous upper and lower bounds, our results characterize, up to logarithmic terms, the accuracy of graphon estimation in the $$\delta_2$$ metric. A similar lower bound to ours was obtained independently by Klopp et al. 1. Introduction Networks and graphs arise as natural modelling tools in many areas of science. In many settings, particularly in social networks, networks display some type of community structure. In these settings, one may consider the nodes of the graph as belonging to one of $$k$$ communities, and two nodes are connected with a probability that depends on the communities they belong to. This type of structure is captured in the $$k$$-block graphon model, also known as stochastic block models. The more communities we allow in the model (or ‘types’ of nodes we consider), the richer the model becomes and the better we can hope to describe the real world. One can think of a general graphon model as an $$\infty$$-block graphon, where each node is given a label in $$[0,1]$$ rather than $$\{1, \ldots, k\}$$. Given an observed network, graphon estimation is the problem of recovering the graphon model from which the graph was drawn. In this article, we are concerned with the fundamental limits of graphon estimation for block graphons. That is, given an $$n$$-node network that was generated from a $$k$$-block graphon, how accurately can you recover the graphon? We consider the ‘non-parametric’ setting, where $$k$$ may depend on $$n$$. Our lower bounds apply even to estimation algorithms that know the true number of blocks $$k$$, though this quantity typically needs to be estimated. In many real-world networks, the average degree of the network is small compared with the number of nodes in the network. Graphons whose expected average degree is linear in $$n$$ are called dense, while graphons whose expected average degree is sublinear in $$n$$ are referred to as sparse. In this work, we prove a new lower bound for graphon estimation for sparse networks. In particular, our results rule out non-trivial estimation for very sparse networks (roughly, where $$\rho = O(k^2/n^2)$$). An estimator is non-trivial if its expected error is significantly better than an estimator that ignores the input and always outputs the same model. It follows from recent work [1–3] that non-trivial estimation is impossible when $$\rho=O(1/n)$$. Ours is the first lower bound that rules out non-trivial graphon estimation for large $$k$$. Previous work by Klopp et al. [4] provides other lower bounds on graphon estimation that are tight in several regimes. In recent work [5] that is concurrent to ours, the same authors provide a similar bound to the one presented here. Block graphon models were introduced by Hoff et al. [6] under the name latent position graphs. Graphons play an important role in the theory of graph limits (see [7] for a survey) and the connection between the graph model, and convergent graph sequences has been studied in both the dense and the sparse settings [8–11]. Estimation for stochastic block models with a fixed number of blocks was introduced by Bickel and Chen [12], while the first estimation of the general model was proposed by Bickel et al. [13]. Since then, many graphon estimation methods, with an array of assumptions on the graphon, have been proposed [14–24]. Gao et al. [19] provide the best known upper bounds in the dense setting, while Wolfe and Olhede [23], Borgs et al. [25] and Klopp et al. [4] give upper bounds for the sparse case. 1.1 Graphons Definition 1 (Bounded graphons and $$W$$-random graphs) A (bounded) graphon$$W$$ is a symmetric, measurable function $$W:[0,1]^2\to [0,1]$$. Here, symmetric means that $$W(x,y)=W(y,x)$$ for all $$(x,y)\in [0,1]^2$$. For any integer $$n$$, a graphon $$W$$ defines a distribution on graphs on $$n$$ vertices as follows: Firstly, select $$n$$ labels $$\ell_1, \ldots, \ell_n$$ uniformly and independently from $$[0,1]$$ and form an $$n\times n$$ matrix $$H$$, where $$H_{ij} = W(\ell_i, \ell_j)$$. We obtain an unlabelled, undirected graph $$G$$ by connecting the $$i$$th and $$j$$th nodes with probability $$H_{ij}$$ independently for each $$(i,j)$$. The resulting random variable is called a $$W$$-random graph and denoted $$G_n(W)$$. For $$\rho \geq 0$$, we say a graphon is $$\rho$$-bounded if $$W$$ takes values in $$[0,\rho]$$ (that is, $$\|W\|_\infty \leq \rho$$). We denote the set of graphs with $$n$$ nodes by $$\mathcal{G}_n$$, the set of graphons by $$\mathcal{W}$$ and the set of $$\rho$$-bounded graphons by $$\mathcal{W}_\rho$$. If $$W$$ is $$\rho$$-bounded, then the expected number of edges in $$G_n(W)$$ is at most $$\rho\binom n 2 = O(\rho n^2)$$. In the case that $$\rho$$ depends on $$n$$ and $$\lim_{n\to\infty}\rho\to0$$, we obtain a sparse graphon. We consider the estimation problem: given parameters $$n$$ and $$\rho$$, as well as a graph $$G\sim G_n(W)$$ generated from an unknown $$\rho$$-bounded graphon $$W$$, how well can we estimate $$W$$? A natural goal is to design estimators that produce a graphon $$\hat W$$ that is close to $$W$$ in a metric such as $$L_2$$. This is not possible, because there are many graphons that are far apart in $$L_2$$, but that generate the same probability distribution on graphs. If there exists a measure-preserving map $$\phi:[0,1]\to[0,1]$$, such that $$W(\phi(x), \phi(y))=W'(x,y)$$ for all $$x,y\in[0,1]$$, then $$G_n(W)$$ and $$G_n(W')$$ are identically distributed. The converse is true if we instead only require $$W(\phi(x), \phi(y))=W'(x,y)$$ almost everywhere. Thus, we wish to say that $$\hat W$$ approaches the class of graphons that generate $$G_n(W)$$. To this end, we use the following metric on the set of graphons, \begin{equation} \delta_2(W,W') = \inf_{{\phi:[0,1]\to[0,1] \atop \text{measure-preserving}}} {\left\| {{W_{\phi}-W'}} \right\|_2}\!, \end{equation} (1.1) where $$W_{\phi}(x,y)=W(\phi(x),\phi(y))$$ and $$\phi$$ ranges over all measurable, measure-preserving maps. Two graphons $$W$$ and $$W'$$ generate the same probability distribution on the set of graphs if and only if $$\delta_2(W, W')=0$$ (see Lovász [7], for example). Existing upper bounds for graphon estimation are based on algorithms that produce graphons of a particular form, namely block graphons, also called stochastic block models (even when it is not known that the true graphon is a block graphon). Definition 2 ($$k$$-block graphon (stochastic block models)) For $$k\in\mathbb{N}$$, a graphon is a $$k$$-block graphon if there exists a partition of $$[0,1]$$ into $$k$$ measurable sets $$I_1, \ldots, I_k$$, such that $$W$$ is constant on $$I_i\times I_j$$ for all $$i$$ and $$j$$. We can associate a graphon of this form to every square matrix. Given a $$k\times k$$ symmetric matrix $$M$$, let $$W[M]$$ denote the $$k$$-block graphon with blocks $$I_i=\big(\frac{i-1}{k}, \frac{i}{k}\big]$$ that takes the value $$M_{ij}$$ on $$I_i\times I_j$$. 1.2 Main result We are concerned with the problem of estimating a graphon, $$W$$, given a graph sampled from $$G_n(W)$$. A graphon estimator is a function $$\hat{W}: \mathcal{G}_n \to \mathcal{W}$$ that takes as input an $$n$$ node graph, that is generated according to $$W$$, and attempts to output a graphon that is close to $$W$$. The main contribution of this article is the development of the lower bound \begin{equation} \inf_{\hat{W}}\sup_{W}\underset{G\sim G_n(W)}{\mathbb{E}}[\delta_2(\hat{W}(G), W)]\ge{\it{\Omega}}\left(\min\left(\rho, \sqrt{\frac{\rho k^2}{n^2}}\right)\right)\!. \end{equation} (1.2) Combined with previous work, we can give the following lower bound on the error of graphon estimators. Theorem 3 For any positive integer $$2\le k\le n$$ and $$0< \rho\le 1$$, \begin{equation} \inf_{\hat{W}}\sup_{W}\underset{G\sim G_n(W)}{\mathbb{E}}\left[\delta_2(\hat{W}(G), W)\right]\ge {\it{\Omega}}\left(\min\left(\rho, \;\rho\sqrt[4]{\frac{k}{n}}+\sqrt{\frac{\rho k^2}{n^2}}\right)\right)\!, \end{equation} (1.3) where $$\inf_{\hat{W}}$$ is the infimum over all estimators $$\hat{W}:G_n\to\mathcal{G}$$ and $$\sup_W$$ is the supremum over all $$k$$-block, $$\rho$$-bounded graphons. If $$\rho n$$ is non-decreasing and there exists a constant $$c>0$$, such that $$\rho n\ge c$$, then $$\inf_{\hat{W}}\sup_{W}\underset{G\sim G_n(W)}{\mathbb{E}}\left[\delta_2(\hat{W}(G), W)\right]\ge {\it{\Omega}}\left(\min\left(\rho, \;\rho\sqrt[4]{\frac{k}{n}}+\sqrt{\frac{\rho k^2}{n^2}}+\sqrt{\frac{\rho}{n}}\right)\right)\!.$$ Note that $$k$$ and $$\rho$$ may depend on $$n$$. That is, the theorem holds if we consider sequences $$\rho_n$$ and $$k_n$$. Our result improves on previously known results when $$\rho=O\left(\left(\frac{k}{n}\right)^{3/2}\right)$$—that is, when the graphs produced by the graphon are sparse. The upper bound \begin{equation}\label{Kloppupper} \inf_{\hat{W}}\sup_{W}\underset{G\sim G_n(W)}{\mathbb{E}}\left[\delta_2(\hat{W}(G), W)\right]\le O\left(\min\left(\rho, \;\rho\sqrt[4]{\frac{k}{n}}+\sqrt{\frac{\rho k^2}{n^2}}+\sqrt{\frac{\rho \log k}{n}}\right)\right) \end{equation} (1.4) by Klopp et al. [4] implies that our lower bound is almost tight. In particular, if $$k$$ is constant and $$\rho$$ is within the designated range then the lower bound in Theorem 3 is tight. When $$\rho=O\left(\frac{k^2}{n^2}\right)$$, Theorem 3 implies that the error is $${\it{\Omega}}(\rho)$$, which is the error achieved by the trivial estimator $$\hat{W}=0$$. That is, in the sparse setting, the trivial estimator achieves the optimal error. To the authors’ knowledge, this is the first result that completely rules out non-trivial estimation in the case where $$k$$ is large. Recent concurrent work ([5]) provides similar bounds. The bound \begin{equation}\label{Klopplower} \inf_{\hat{W}}\sup_{W}\underset{G\sim G_n(W)}{\mathbb{E}}[\delta_2(\hat{W}, W)]\ge {\it{\Omega}}\left(\rho\sqrt[4]{\frac{k}{n}}\right) \end{equation} (1.5) is due to previous work of Klopp et al. [4] and the bound \begin{equation}\label{Neemanlower} \inf_{\hat{W}}\sup_{W}\underset{G\sim G_n(W)}{\mathbb{E}}[\delta_2(\hat{W}, W)]\ge{\it{\Omega}}\left(\min\left(\rho, \sqrt{\frac{\rho}{n}}\right)\right) \end{equation} (1.6) for constant $$k$$ follows from the results of Mossel et al. [1] and Banerjee [26]. We give details on how to derive (1.6) from their results in the Appendix. 1.3 Techniques: combinatorial lower bounds for $$\delta_p$$ Our proof of the main theorem will involve Fano’s lemma. As such, during the course of the proof, we will need to lower bound the packing number, with respect to $$\delta_2$$, of a large set of $$k$$-block graphons. While easily upper bounded, little is known about lower bounds on $$\delta_2$$. To the authors’ knowledge, this work gives the first lower bound for the packing number of $$\mathcal{W}_\rho$$ with respect to $$\delta_2$$. We will also give a combinatorial lower bound for the $$\delta_2$$ metric that is easier to handle than the metric itself. To understand our technical contributions, it helps to first understand a problem related to graphon estimation, namely that of estimating the matrix of probabilities $$H$$. Existing algorithms for graphon estimation are generally analysed in two phases: firstly, one shows that the estimator $$\hat W$$ is close to the matrix $$H$$ (in an appropriate version of the $$\delta_2$$ metric) and then uses (high probability) bounds on $$\delta_2(W,W[H])$$ to conclude that $$\hat{W}$$ is close to $$W$$. Klopp et al. [4] show tight upper and lower bounds on estimation of $$H$$. One can think of our lower bound as showing that the lower bounds on estimation of $$H$$ can be transferred to the problem of estimating $$W$$. The main technical difficulty lies in showing that a given pair of matrices $$A,B$$ lead to graphons that are far apart in the $$\delta_2$$ metric. Even if $$A,B$$ are far apart in, say, $$\ell_2$$, they may lead to graphons that are close in $$\delta_2$$. For consistency with the graphon formalism, we normalize the $$\ell_2$$ metric on $$k\times k$$ matrices, so that it agrees with the $$L_2$$ metric on the corresponding graphons. For a $$k\times k$$ matrix $$A$$, \begin{equation} {\left\| {A} \right\|_2} {\overset{\text{def}}{=}} \Big(\frac 1 {k^2} \sum_{i,j\in [k]} A_{ij}^2\Big)^{1/2} = {\left\| {{W[A]}} \right\|_2}\!. \end{equation} (1.7) As an example of the discrepancy between the $$\ell_2$$ and $$\delta_2$$ metrics, consider the matrices $$A = \begin{pmatrix} 1 & 0 & 1\\ 0&1&0\\1&0&1\end{pmatrix} \hspace{0.5in}\text{and}\hspace{0.5in} B= \begin{pmatrix} 1 & 1 & 0\\ 1&1&0\\0&0&1\end{pmatrix}.$$ The matrices $$A$$ and $$B$$ have positive distance in the $$\ell_2$$ metric, $${\left\| {{A-B}} \right\|_2} = \frac{2}{3}$$, but $$\delta_2(W[A], W[B])=0$$. One can get an upper bound on $$\delta_2(W[A],W[B])$$ by restricting attention in the definition of $$\delta_2$$ to functions $$\phi$$ that permute the blocks $$I_i$$. This leads to the following metric on $$k\times k$$ matrices, which minimizes over permutations of the rows and columns of one of the matrices: \begin{equation} \hat{\delta}_2(A,B) {\overset{\text{def}}{=}} \min_{\sigma\in\mathcal{S}_k}{\left\| {{A_{\sigma}-B}} \right\|_2}\!, \end{equation} (1.8) where $$A_\sigma$$ is the matrix with entries $$(A_\sigma)_{ij} = A_{\sigma(i),\sigma(j)}$$. This metric arises in other work (e.g. [7]), and it is well known that \begin{equation} \delta_2(W[A],W[B]) \leq \hat{\delta}_2(A,B). \end{equation} (1.9) To prove lower bounds, we consider a new metric on matrices, in which we allow the rows and columns to be permuted separately. Specifically, let \begin{equation} \hat{\hat{{\delta}}}_2(A,B) {\overset{\text{def}}{=}} \min_{\sigma, \tau\in\mathcal{S}_k}{\left\| {{A_{\sigma, \tau}-B}} \right\|_2}\!, \end{equation} (1.10) where $$A_{\sigma,\tau}$$ is the $$k\times k$$ matrix with entries $$(A_{\sigma,\tau})_{ij} = A_{\sigma(i),\tau(j)}$$. Lemma 1 (Lower bound for $$\delta_2$$) For every two $$k\times k$$ matrices $$A,B$$, \begin{equation} \hat{\hat{\delta_2(A,B)}} \leq \delta_2(W[A], W[B]). \end{equation} (1.11) Because $$\hat{\hat{\delta_2}}$$ is defined ‘combinatorially’ (that is, it involves minimization over a discrete set of size about $$2^{2k\ln k}$$, instead of over all measure-preserving injections), it is fairly easy to lower bound $$\hat{\hat{ \delta_2(A,B)}}$$ for random matrices $$A,B$$ using the union bound. In particular, it allows us to give bounds on the packing number of $$\mathcal{W}_\rho$$ with respect to the $$\delta_2$$ metric. If $$({\it{\Omega}}, d)$$ is a metric space, $$\epsilon>0$$ and $$T\subset{\it{\Omega}}$$, then we define the $$\epsilon$$-packing number of $$T$$ to be the largest number of disjoint balls of radius $$\epsilon$$ that can fit in $$T$$, denoted by $$\mathcal{M}(\epsilon, T, d)$$. The following proposition will be proved after the proof of Theorem 3. Proposition 1 There exists $$C>0$$ such that the $$C\rho$$-packing number of $$\mathcal{W}_{\rho}$$, equipped with $$\delta_2$$, is $$2^{{\it{\Omega}}(k^2)}$$, that is $$\mathcal{M}(C\rho, \mathcal{W}_{\rho}, \delta_2) = 2^{{\it{\Omega}}(k^2)}$$. Finally, we note that these techniques extend directly to the $$\delta_p$$ metric, for $$p \in [1,\infty]$$. That is, we may define $$\delta_p, \hat\delta_p$$ and $$\hat{\hat{\delta_p}}$$ analogously to the definitions above and obtain the bounds \begin{equation} \hat{\hat{\delta_p(A,B)}} \leq \delta_p(W[A], W[B])\leq \hat\delta_p(A,B), \end{equation} (1.12) along with similar lower bounds on the packing number. 1.4 Related work Work on graphon estimation falls broadly into two categories: estimating the matrix $$H$$ and estimating the graphon $$W$$. When estimating $$H$$, the aim is to produce a matrix that is close in the $$\ell_2$$ metric to the true matrix of probabilities $$H$$ that was used to generate the graph $$G$$. When estimating the graphon, our aim is to minimize the $$\delta_2$$ distance between the estimate and the true underlying graphon $$W$$ that was used to generate $$G$$. Gao et al. [19] studied the problem of estimating the matrix of probabilities $$H$$, given an instance chosen from $$W$$ when $$\rho=1$$. They proved the following minimax rate for this problem when $$W$$ is a $$k$$-block graphon: \begin{equation} \inf_{\hat{M}}\sup_{H}\underset{G\sim G_n(H)}{\mathbb{E}}\left[\frac{1}{n^2}{\left\| {{\hat{M}(G)-H}} \right\|_2}\right]\asymp \sqrt{\frac{k^2}{n^2}+\frac{\log k}{n}}, \end{equation} (1.13) where the infinimum is over all estimators $$\hat{M}$$ from $$G_n$$ to the set of symmetric $$n\times n$$ matrices, the supremum is over all probability matrices $$H$$ generated from $$k$$-block graphons. Klopp et al. [4] extended this result to the sparse case, proving that for all $$k\le n$$ and $$0<\rho\le 1$$, \begin{equation} \inf_{\hat{M}}\sup_{H}\underset{G\sim G_N(H)}{\mathbb{E}}\left[\frac{1}{n^2}{\left\| {{\hat{M}(G)-H}} \right\|_2}\right]\ge{\it{\Omega}}\left(\min\left(\sqrt{\rho\left(\frac{k^2}{n^2}+\frac{\log k}{n}\right)}, \rho\right)\right)\!, \end{equation} (1.14) where the supremum is over all probability matrices $$H$$ generated from $$k$$-block, $$\rho$$-bounded graphons. Klopp et al. [4, Corollary 3] also studied the problem of estimating the graphon $$W$$. They proved that equation (1.4) holds for any $$k$$-block, $$\rho$$-bounded graphon, $$W$$, with $$k\le n$$. They also exhibited the first lower bound (known to us) for graphon estimation using the $$\delta_2$$ metric. They proved that equation (1.5) holds for $$\rho>0$$ and $$k\le n$$. The related problems of distinguishing a graphon with $$k> 1$$ from an Erdös–Rényi model with the same average degree (called the distinguishability problem) and reconstructing the communities of a given network (called the reconstruction problem) have also been widely studied. This problem is closely related to the problem of estimating $$H$$. Recent work by Mossel et al. [1] and Neeman and Netrapalli [3] establish conditions under which a $$k$$-block graphon is mutually contiguous to the Erdös–Rényi model with the same average degree. Contiguity essentially implies that no test could ever definitely determine which of the two graphons a given sample came from. There is a large body of work on algorithmic and statistical problems in this area, and we have only cited work that is directly relevant here. 2. Lower bound for the $$\delta_2$$ metric As mentioned earlier, the main technical contribution of this article is lower bounding the $$\delta_2$$ metric by the more combinatorial $$\hat{\hat{{\delta_2}}}$$ metric. In this section, we will prove the inequality given in Lemma 1. Proposition 2 Let $$W, W'$$ be $$k$$-block graphons with blocks $$I_i=\big[\frac{i-1}{k}, \frac{i}{k}\big)$$ and $$\pi:[0,1]\to[0,1]$$ be a measure-preserving map. Then there exists a probability distribution $$\mathbb{P}$$ on $$\mathcal{S}_k$$ such that \begin{equation} {\left\| {{W_{\pi}-W'}} \right\|_2}^2 = \underset{\sigma, \tau\sim\mathbb{P}}{\mathbb{E}}\left[{\left\| {{W_{\sigma, \tau}-W'}} \right\|_2}^2\right], \end{equation} (2.1) where the expectation is taken over $$\sigma,\tau$$ selected independently according to $$\mathbb{P}$$. Proof. Let $$a_i = \frac{i-1}{k}$$ and $$p_{ij} = \mu(I_i\cap \pi^{-1}(I_j))$$. Now, consider a $$k\times k$$ matrix $$P$$ with $$P_{ij} = kp_{ij}$$. Noting that $$\sum_{j=1}^k p_{ij} = \mu(I_i) = 1/k$$ and $$\sum_{i=1}^k p_{ij} = \mu(\pi^{-1}(I_j)) = 1/k$$, we can see that $$P$$ is doubly stochastic, that is, the rows and columns of $$P$$ sum to 1. Berkhoff’s theorem states that any doubly stochastic matrix can be written as a convex combination of permutation matrices. Therefore, we have a probability distribution $$\mathbb{P}$$ on $$\mathcal{S}_k$$ such that $$P=\sum_{\sigma\in\mathcal{S}_k}\mathbb{P}(\sigma)\sigma$$ and $$\sum_{\sigma\in\mathcal{S}_k}\mathbb{P}(\sigma)=1$$ and \begin{equation} \mathbb{P}(\sigma(i)=j) = \sum\{\mathbb{P}(\sigma) \:|\: \sigma(i)=j\} = P_{ij} = kp_{ij}. \end{equation} (2.2) Taking expectations over $$\sigma,\tau$$ selected independently from $$\mathbb{P}$$, \begin{align*} \mathbb{E}\left[{\left\| {{W_{\sigma, \tau}-W'}} \right\|_2}^2\right]&=\sum_{\sigma,\tau} \mathbb{P}(\sigma)\mathbb{P}(\tau)\sum_{i,j} \frac{1}{k^2}(W(a_{\sigma(i)}, a_{\tau(j)})-W'(a_i,a_j))^2 \\ &= \sum_{i,i',j,j'} \frac{1}{k^2}\mathbb{P}(\sigma(i)=i')\mathbb{P}(\tau(j)=j')(W(a_i, a_j)-W'(a_{i'}, a_{j'}))^2 \\ &= \sum_{i,i',j,j'} p_{ii'}p_{jj'}(W(a_i, a_j)-W'(a_{i'},a_{j'}))^2\\ &= {\left\| {{W_{\pi}-W'}} \right\|_2}^2\!. \end{align*} □ Proof of Lemma 1. Proposition 2 implies that for all measure-preserving maps $$\pi:[0,1]\to[0,1]$$ and matrices $$A$$ and $$B$$ we have \begin{equation} {\left\| {{W[A]_{\pi}-W[B]}} \right\|_2}\ge\inf_{\sigma, \tau\in\mathcal{S}_k}{\left\| {{W[A]_{\sigma, \tau}-W[B]}} \right\|_2}=\inf_{\sigma, \tau\in\mathcal{S}_k}{\left\| {{A_{\sigma, \tau}-B}} \right\|_2} = \hat{\hat{{\delta_2}}}(A, B). \end{equation} (2.3) Since this is true for any $$\pi$$, we have $$\delta_2(W[A],W[B])\ge\hat{\hat{{\delta_2}}}(A,B)$$. □ 3. Proof of main theorem To prove the main theorem, we will use Fano’s lemma to find a constant that lower bounds the probability that the estimation exceeds $$\min\left(\rho, \sqrt{\frac{\rho k^2}{n^2}}\right)$$, which then implies the appropriate lower bound on the expected $$\delta_2$$ error. To that end, we aim to find a large set, $$T$$, of $$k$$-block graphons whose Kullback–Leibler (KL) diameter and $$\epsilon$$-packing number with respect to $$\delta_2$$ with $$\epsilon=\min\left(\rho, \sqrt{\frac{\rho k^2}{n^2}}\right)$$ can be bounded. Our proof is inspired by that of Gao et al. [19]. Suppose $$p, q$$ are probability distributions on the same space. Then the (KL) divergence of $$p$$ and $$q$$ is defined by $$D(p\|q) = \int\big(\log\frac{{\rm{d}}p}{{\rm{d}}q}\big){\rm{d}}p$$. For a collection $$T$$ of probability distributions, the KL diameter is defined by \begin{equation} {\rm d}_{\rm KL}(T)=\sup_{p,q\in T} D(p\| q). \end{equation} (3.1) The following version of Fano’s lemma is found in [27]. Recall that $${\log\mathcal{M}(\epsilon, T, d)}$$ is the size of the largest $$\epsilon$$-packing of $$T$$ (with distance $$d$$). Lemma 2 (Fano’s inequality) Let $$({\it{\Omega}}, d)$$ be a metric space and $$\{\mathbb{P}_{\theta} \: |\: \theta\in{\it{\Omega}}\}$$ be a collection of probability measures. For any totally bounded $$T\subset{\it{\Omega}}$$ and $$\epsilon>0$$, \begin{equation} \inf_{\hat{\theta}}\sup_{\theta\in{\it{\Omega}}}\mathbb{P}_{\theta}\left({\rm d}^2(\hat{\theta}(X), \theta)\ge \frac{\epsilon^2}{4}\right)\ge 1-\frac{{\rm d}_{\rm KL}(P_T)+1}{\log\mathcal{M}(\epsilon, T, d)}, \end{equation} (3.2) where the infimum is over all estimators and $$P_T = \{\mathbb{P}_{\theta}\;|\;\theta\in T\}$$. The following lemma gives us a way to easily upper bound the KL divergence between the distributions induced by two different graphons. Lemma 3 For any graphons $$\frac{1}{4}\le W,W'\le \frac{3}{4}$$, we have \begin{equation} D(G_n(W)\|G_n(W'))\le 8n^2{\left\| {W-W'} \right\|_2}^2\!. \end{equation} (3.3) Proof. Let $$T$$ be a variable denoting the choice of labels, so \begin{equation} \mathbb{P}_{G_n(W)}(G) = \int_{\ell\in[0,1]^n} \mathbb{P}_T(\ell)\mathbb{P}_{G_n(W)}(G|T=\ell)\,{\rm{d}}\ell. \end{equation} (3.4) Now, \begin{align*} D(G_n(W)\|G_n(W')) &= \sum_{G\in G_n} \mathbb{P}_{G_n(W)}(G)\ln\left(\frac{\mathbb{P}_{G_n(W)}(G)}{\mathbb{P}_{G_n(W')}(G)}\right)\\ &\le \sum_{G\in G_n} \int_{\ell\in[0,1]^n} \mathbb{P}_T(\ell)\mathbb{P}_{G_n(W)}(G|T=\ell)\ln\left(\frac{\mathbb{P}_{G_n(W)}(G|T=\ell)}{\mathbb{P}_{G_n(W')}(G|T=\ell)}\right){\,{\rm d}}\ell\\ &= \int_{\ell\in[0,1]^n} \mathbb{P}_T(\ell)D(\mathbb{P}_{G_n(W)}(\cdot|T=\ell))\|\mathbb{P}_{G_n(W')}(\cdot|T=\ell){\, {\rm d}}\ell, \end{align*} where the inequality follows from the log-integral inequality [28, Theorem 30.12.4]. Now, the probability density function of $$T$$ is the constant function 1, so it follows from Gao et al. [19, Proposition 4.2] that \begin{align*} D(G_n(W)\|G_n(W')) &\le 8\int_{\ell\in[0,1]^n}\sum_{i,j=1}^n (W({\ell_i},{\ell_j})- W'({\ell_i},{\ell_j}))^2{\, {\rm d}}\ell\\ &=8\sum_{i,j=1}^n \int_{\ell\in[0,1]^n} (W(\ell_i,\ell_j)- W'(\ell_i,\ell_j))^2 {\, {\rm d}}\ell\\ &\le 8n^2\int_{[0,1]^2} (W(x,y)- W'(x,y))^2{\, {\rm d}}x\,{\rm{d}}y\\ &= 8n^2{\left\| {W-W'} \right\|_2}^2\!. \end{align*} □ Recall that we are aiming to define a large set of $$k$$-block matrices that are close in KL divergence, but that are $$\epsilon$$-far apart with respect to $$\delta_2$$ (with $$\epsilon=\min(\rho, \sqrt{\frac{\rho k^2}{n^2}})$$). The following lemma shows that there exists a large set of matrices that are pairwise far in Hamming distance, even after every possible permutation of the rows and columns. We will use this in the proof of Theorem 3 to define a large class of $$k$$-block graphons that are pairwise far in the $$\hat{\hat{{\delta_2}}}$$ metric and hence the $$\delta_2$$ metric. This gives us a bound on packing number. Lemma 4 There exists a set $$S$$ of symmetric $$k\times k$$ binary matrices such that $$|S| = 2^{{\it{\Omega}}(k^2)}$$ and, for every $$B, B'\in S$$ and $$\sigma, \tau\in\mathcal{S}_k$$, we have Ham$$(B_{\sigma, \tau}, B')= {\it{\Omega}}(k^2)$$. Proof. Fix permutations $$\sigma$$ and $$\tau$$, and consider two randomly chosen symmetric binary matrices $$B, B'$$. For $$i\le j$$, let $$X_{ij}=1$$ if $$B_{\sigma(i), \tau(j)}=B'_{i,j}$$ and 0 otherwise so $$X_{ij}$$ is a Bernoulli random variable with $$\mathbb{E}[X_{ij}]=\frac{1}{2}$$. Thus, by a Chernoff bound, \begin{equation} \mathbb{P}\left(\text{Ham}(B_{\sigma, \tau}, B')<\frac{k^2}{6}\right)=\mathbb{P}\left(\sum_{i\le j}X_{ij}\le \frac{k^2}{6}\right)\le e^{\frac{-2\left(\frac{k^2}{6}-\frac{1}{2}\binom{k}{2}\right)^2}{\binom{k}{2}}}. \end{equation} (3.5) Therefore, for randomly chosen $$B, B'$$, \begin{equation} \mathbb{P}\left(\exists\sigma, \tau \text{ s.t. Ham}(B_{\sigma, \tau}, B')<\frac{k^2}{6}\right)\le e^{\frac{-2\left(\frac{k^2}{6}-\frac{1}{2}\binom{k}{2}\right)^2}{\binom{k}{2}}}(k!)^2 = 2^{-{\it{\Omega}}(k^2)}. \end{equation} (3.6) For a constant $$c>0$$, consider the process that selects $$2^{ck^2}$$ binary matrices $$\{B_i\}_{i}$$ uniformly at random uniformly at random. The probability that all pairs are at Hamming distance at least $$k^2/6$$ is at least $$1-2^{2ck^2}2^{-{\it{\Omega}}(k^2)}$$. Selecting $$c$$ sufficiently small, we get that at least one such set $$S$$ exists. □ We are not aware of an explicit construction of a large family of matrices that are far apart in $$\hat{\hat{{\delta}_2}}$$ metric; we leave such a construction as an open problem. We now proceed to the proof of Theorem 3. We will use Lemma 4 to define a set $$T$$ with packing number $$2^{{\it{\Omega}}(k^2)}$$. The elements of $$T$$ are all close in $$\|\cdot\|_{\infty}$$ norm, so using Lemma 3 we get a bound on the KL diameter. We then directly apply these bounds via Fano’s lemma. Theorem 4 For any positive integer $$k\le n$$ and $$0< \rho\le 1$$, \begin{equation} \inf_{\hat{W}}\sup_{W}\underset{G\sim G_n(W)}{\mathbb{E}}\left[\delta_2(\hat{W}(G), W)\right]\ge {\it{\Omega}}\left(\min\left(\rho, \;\sqrt{\frac{\rho k^2}{n^2}}\right)\right)\!, \end{equation} (3.7) where $$\inf_{\hat{W}}$$ is the infimum over all estimators $$\hat{W}:G_n\to\mathcal{G}$$ and $$\sup_W$$ is the supremum over all $$k$$-block, $$\rho$$-bounded graphons. Proof. Let $$S$$ be a set satisfying the conditions of Lemma 4 and let $$\eta = \min(1, \frac{k}{n\sqrt{\rho}})$$. For $$B\in S$$, define \begin{equation} Q_B = \rho\left[\frac{1}{2}\mathbf{1}+c\eta(2B-\mathbf{1})\right]\!, \end{equation} (3.8) where $$\mathbf{1}$$ is the all 1’s matrix and $$c$$ is some constant that we will choose later. That is, $$(Q_B)_{ij}=\rho[\frac{1}{2}+c\eta]$$ if $$B_{ij}=1$$ and $$(Q_B)_{ij} = \rho[\frac{1}{2}-c\eta]$$ if $$B_{ij}=0$$. Let $$T= \left\{W[Q_B] \:|\: B\in S\right\}$$. Using Lemma 3, we conclude that for all $$W, W'\in T$$, we have \begin{equation} D(G_n(W)\|G_n(W')) \le 8n^2(2c\rho\eta)^2 \le 32c^2k^2\rho, \end{equation} (3.9) so $${\rm d}_{\rm KL}(T) = O(c^2k^2\rho)$$. Let $$B, B'\in S$$ and suppose $$\sigma, \tau\in\mathcal{S}_k$$. By construction, \begin{equation} {\left\| {{({W[Q_B]})_{\sigma, \tau}-W[Q_{B'}]}} \right\|_2}^2 \ge \frac{1}{k^2}\text{Ham}(B_{\sigma, \tau}, B')(2\rho c\eta)^2 = {\it{\Omega}}(c^2\rho^2\eta^2). \end{equation} (3.10) Thus by Corollary 2, \begin{equation} \delta_2(W[Q_B], W[Q_{B'}])\ge\hat{\hat{{\delta_2}}}(W[Q_B], W[Q_{B'}])\ge {\it{\Omega}}(c\rho\eta). \end{equation} (3.11) Therefore, there exists $$D>0$$ such that if $$\epsilon = D\rho c\eta = D\min\left(c\rho, \frac{ck\sqrt{\rho}}{n}\right)$$, we have $$\log\mathcal{M}(\epsilon, T, \delta_2) = {\it{\Omega}}(k^2)$$. Fano’s lemma implies \begin{equation} \inf_{\hat{W}}\sup_{W} \text{Pr}\left(\delta_2(\hat{W}, W)\ge \frac{\epsilon}{2}\right)\ge 1-\frac{O(c^2k^2\rho)+1}{{\it{\Omega}}(k^2)}. \end{equation} (3.12) We can choose $$c$$ small enough that the right-hand side is larger than a fixed constant for all $$k$$ and $$n$$. By Markov’s inequality, we have \begin{equation} \inf_{\hat{W}}\sup_{W}\mathbb{E}\left[\delta_2(\hat{W}, W)\right] = {\it{\Omega}}\left(\epsilon\right) = {\it{\Omega}}\left(\min\left(\rho, \sqrt{\rho\frac{k^2}{n^2}}\right)\right)\!. \end{equation} (3.13) □ Proof of Proposition 1. During the course of the proof of Theorem 3, we construct $$2^{{\it{\Omega}}(k^2)}$$ graphons in $$\mathcal{W}_{\rho}$$ that are pairwise at least $${\it{\Omega}}(\rho c\eta)$$ apart in the $$\delta_2$$ distance for any $$c>0$$ such that $$|c\eta|\le\frac{1}{2}$$. Therefore, for some $$C>0$$, the $$C\rho$$-packing number of $$\mathcal{W}_{\rho}$$ is at least $$2^{{\it{\Omega}}(k^2)}$$. □ Acknowledgements We are grateful for helpful conversations with Christian Borgs, Jennifer Chayes, Olga Klopp and Alexandre Tsybakov. Funding Alfred P. Sloan Foundation; a Google Faculty Research Award; and the National Science Foundation award (IIS-1447700). A preliminary version of this work appears as ArXiv report arXiv:1604.01871 [math.ST]. This work was done while the author was visiting Pennsylvania State University. References 1. Mossel E. , Neeman J. & Sly A. ( 2014 ) Reconstruction and estimation in the planted partition model . Probab. Theory Related Fields , 162 , 431 – 461 . Google Scholar CrossRef Search ADS 2. Mossel E. , Neeman J. & Sly A. ( 2015 ) Consistency thresholds for the planted bisection model . Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC ’15 ( Rubinfeld R. ed.). 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T. , Cohn H. & Zhao Y. ( 2014 ) An $${L}^p$$ theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions . arXiv:1401.2906 . 9. Borgs C. , Chayes J. T. , Cohn H. & Zhao Y. ( 2014 ) An $${L}^p$$ theory of sparse graph convergence II: LD convergence, quotients, and right convergence . arXiv:1408.0744 . 10. Borgs C. , Chayes J. T. , Lovász L. , Sós V. & Vesztergombi K. ( 2006 ) Counting graph homomorphisms . Topics in Discrete Mathematics ( Klazar M. Kratochvil J. Loebl M. Matousek J. Thomas R. & Valtr P. eds). Berlin, Heidelberg : Springer , pp. 315 – 371 . Google Scholar CrossRef Search ADS 11. Borgs C. , Chayes J. T. , Lovász L. , Sós V. & Vesztergombi K. ( 2008 ) Convergent graph sequences I: subgraph frequencies, metric properties, and testing . Adv. Math. , 219 , 1801 – 1851 . Google Scholar CrossRef Search ADS 12. Bickel P. J. & Chen A. ( 2009 ) A nonparametric view of network models and Newman–Girvan and other modularities . Proc. Natl. 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Lake Tahoe, USA : Curran Associates, Inc. , pp. 998 – 1006 . 22. Tang M. , Sussman D. L. & Priebe C. E. ( 2013 ) Universally consistent vertex classification for latent positions graphs . Ann. Statist. , 41 , 1406 – 1430 . Google Scholar CrossRef Search ADS 23. Wolfe P. & Olhede S. C. ( 2013 ) Nonparametric graphon estimation . arXiv:1309.5936 . 24. Yang J. J. , Han Q. & Airoldi E. M. ( 2014 ) Nonparametric estimation and testing of exchangeable graph models . Proceedings of 17th International Conference on Artificial Intelligence and Statistics ( Kaski S. & Corander J. eds). Reykjavik, Iceland : PMLR , pp. 1060 – 1067 . 25. Borgs C. , Chayes J. T. & Smith A. ( 2015 ) Private graphon estimation for sparse graphs . Proceedings of the 28th International Conference on Neural Information Processing Systems, NIPS’15 ( Cortes C. Lee D. D. Sugiyama M. & Garnett R. eds). Cambridge, MA : MIT Press , pp. 1369 – 1377 . 26. Banerjee D. ( 2016 ) Contiguity results for planted partition models: the dense case . arXiv:1609.02854v1 . 27. Yu B. ( 1997 ) Assouad, Fano, and Le Cam . Festschrift for Lucien Le Cam ( Pollard D. Torgersen E. & Yang G. eds). New York : Springer , pp. 423 – 435 . Google Scholar CrossRef Search ADS 28. Lapidoth A. ( 2017 ) A Foundation in Digital Communication . Cambridge, UK : Cambridge University Press . Google Scholar CrossRef Search ADS Appendix. Proof of equation (1.6) We show here how to derive the lower bound in (1.6) from the results of Mossel et al. [1] and Banerjee [26]. Let $$({\it{\Omega}}_n, \mathcal{F}_n)$$ be a sequence of measurable spaces, each equipped with two probability measures, $$\mathbb{P}_n$$ and $$\mathbb{Q}_n$$. We say $$\mathbb{P}_n$$ and $$\mathbb{Q}_n$$ are mutually contiguous if for any sequence of events $$A_n$$, we have $$\lim_{n\to\infty}\mathbb{P}_n(A_n)\to0$$ if and only if $$\lim_{n\to\infty}\mathbb{Q}_n(A_n)\to0$$. Lemma A1 Let $$W_1$$ be a $$k$$-block graphon generated by a matrix with diagonal entries, $$p$$, and off-diagonal entries, $$q$$. Let $$W_2$$ be the Erdös–Rényi model with the same expected degree as $$W_1$$. If $$W_1$$ and $$W_2$$ are mutually contiguous then \begin{equation}\label{mosselbound} \inf_{\hat{W}}\sup_{W}\underset{G\sim G_n(W)}{\mathbb{E}}\left[\delta_2(\hat{W}(G), W)\right]\ge {\it{\Omega}}\left(\frac{|p-q|}{\sqrt{k}}\right)\!, \end{equation} (A.1) where $$\inf_{\hat{W}}$$ is the infimum over all estimators $$\hat{W}:G_n\to\mathcal{G}$$ and $$\sup_W$$ is the supremum over all $$k$$-block, $$\max(p,q)$$-bounded graphons. Proof. Let $$\epsilon = |p-q|$$, so $$\delta_2\left(W_1, W_2\right) = \sqrt{\frac{(k-1)^2}{k^3}\epsilon^2+\frac{(k-1)}{k^3}\epsilon^2}\ge C\frac{\epsilon}{\sqrt{k}}$$, for some constant $$C$$. Suppose, for sake of contradiction, that there exists $$\hat{W}$$ such that $$\sup_{W\in\mathcal{W}_{\rho}}\mathbb{E}_{G\sim G_n(W)}\left[\delta_2(\hat{W}(G), W)\right]$$ is not $${\it{\Omega}}\left(\frac{\epsilon}{\sqrt{k}}\right)$$. Then there exists a subsequence $$\{n_t\}_{t\in\mathbb{N}}$$ such that $$\sup_{W\in\mathcal{W}_{\rho}}\mathbb{E}_{G\sim G_{n_t}(W)}\left[\delta_2(\hat{W}(G), W)\right]\le \frac{C}{2t}\frac{\epsilon}{\sqrt{k}}$$ for all $$t\in\mathbb{N}$$. The above inequality, combined with Markov’s inequality, implies \begin{equation}\label{eqn2} \lim_{t\to\infty}Pr_{G\sim G_{n_t}(W_1)}\left[\delta_2(\hat{W}(G), W_1)\ge \frac{C}{2}\frac{\epsilon}{\sqrt{k}}\right]\to 0 \end{equation} (A.2) and \begin{equation}\label{eqn1} \lim_{t\to\infty}Pr_{G\sim G_{n_t}(W_2)}\left[\delta_2(\hat{W}(G), W_2)\ge \frac{C}{2}\frac{\epsilon}{\sqrt{k}}\right]\to 0. \end{equation} (A.3) By equation (A.3) and the contiguity of $$W_1$$ and $$W_2$$, \begin{equation}\label{eqn3} \lim_{t\to\infty}Pr_{G\sim G_{n_t}(W_1)}\left[\delta_2(\hat{W}(G), W_2)\ge \frac{C}{2}\frac{\epsilon}{\sqrt{k}}\right]\to 0. \end{equation} (A.4) Therefore, equations (A.2) and (A.4) imply that for large enough $$n$$, there exists a graph $$G$$ such that $$\delta_2(\hat{W}(G), W_1)< \frac{C}{2}\frac{\epsilon}{\sqrt{k}}$$ and $$\delta_2(\hat{W}(G), W_2)< \frac{C}{2}\frac{\epsilon}{\sqrt{k}}$$, which implies that $$\delta_2(W_1, W_2)<C\frac{\epsilon}{\sqrt{k}}$$, which is a contradiction. □ There are many results in the literature exploring when block graphons are contiguous with the corresponding Erdös–Rényi model. The following table summarizes some of the known results in this area and translates them into lower bounds on the graphon estimation problem via Lemma A1. Let $$\rho=\max(p,q)$$. © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

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Information and Inference: A Journal of the IMAOxford University Press

Published: Aug 23, 2017

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